Non-linear Analysis and Dynamical System

本研究的目的是探讨如何将线性算子理论推广到非线性算子，并利用算子理论、概率论、势理论、微分方程理论等研究非线性理论中的混沌和分形特征。实际上，我们利用算子理论将元胞自动机的极限集分为四种类型：稳定态、循环态、混沌态等。从而将线性算子的一些性质推广到非线性算子。通过将元胞自动机嵌入到有限值上半连续函数空间中，研究了元胞自动机的极限集，定义了元胞自动机收敛于极限集的拓扑，并考虑了线性性质如何推广到非线性的情形，在一定意义下刻画了权函数上强连续半群的动力系统的谱性质.作为另一个刻画，我们利用容许权函数给出了半群是超循环、超循环和混沌的充要条件。由于发展方程的解往往对应于强连续半群，我们可以将这一结果推广到发展方程的性质。有趣的是，即使半群由有界线性算子组成，它有时也会出现混沌。正如已发表的论文列表所示，我们在算子理论、概率论、位势理论、微分方程理论中得到了许多其他的结果，通过考虑如何将线性情况推广到非线性情况，或者考虑什么是非线性分析的本质性质。

The aim of this research is to investigate how we can extend the linear operator theory to non-linear operator and to study chaos and fractal which are characteristic feature in non-linear theory, by using operator theory, probability theory, potential theory, differential equation theory and so on.In fact, we have classified the limit set of cellular automata, by using operator theory, into four types : stable state, cyclic state, chaotic state and others. As a result, we can extend some property of linear operators to non-linear operators. We also investigate the limit set of cellular automata by embedding them into finite-valued upper semi-continuous function spaces and defined the topology in which the cellular automata converge to the limit set, by considering how linear property can be extended to non-linear cases.Dynamical system of strongly continuous semi-groups on weighted functions has already been characterized with their spectral property in some sense. As another characterization, we give necessary and sufficient conditions for the semi-group to be super-cyclic, hyper-cyclic and chaotic, by using admissible weight functions. We can extend this result to the property of evolution equation, since its solution often corresponds to the strongly continuous semi-group. It is interesting that even if the semi-group consists of bounded linear operators, it sometimes occur chaotic. This may happen because the generator is bounded.As shown in the list of published papers, we get many other results in operator theory, probability theory, potential theory, differential equation theory, by considering how we can extend linear case to non-linear case or what is the essential property of non-linear analysis.